In [1]} Amann showed that if \(a\in \widetilde{\mathfrak {M}}_2({\mathcal {L}}(X,Y))\) , then, in particular, the condition (REF ) holds for \(\mathbb {E}\in \lbrace BUC, {\mathcal {C}}_0, L^p; 1\le p\le \infty \rbrace \) . See also [2]}(and the references therein) for a systematic study of the conditions which imply the boundedness of \(a(D)\)  on the classical vector-valued Besov spaces (corresponding to \(L^p\)  spaces).
